The Paradox As Stated By Bertrand
It can be easier to understand the correct answer if you consider the paradox as Bertrand originally described it. After a box has been chosen, but before a drawer is opened to let you observe a coin, the probability is 2/3 that the box has two of the same kind of coin. If the probability of "observing a gold coin" in combination with "the box has two of the same kind of coin" is 1/2, then the probability of "observing a silver coin" in combination with "the box has two of the same kind of coin" must also be 1/2. And if the probability that the box has two like coins changes to 1/2 no matter what kind of coin is shown, the probability would have to be 1/2 even if you hadn't observed a coin this way. Since we know his probability is 2/3, not 1/2, we have an apparent paradox. It can be resolved only by recognizing how the combination of "observing a gold coin" with each possible box can only affect the probability that the box was GS or SS, but not GG.
Read more about this topic: Bertrand's Box Paradox
Famous quotes containing the words paradox and/or stated:
“When a paradox is widely believed, it is no longer recognized as a paradox.”
—Mason Cooley (b. 1927)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)